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Avinas

Fluid Mechanics B.E. (Civil Engineering)

Lecture notes -By : Avinash Pachhai

CHAPTER 1. FLUID AND ITS PROPERTIES 1.1 General introduction 1.1.1 Basics States of matter  Solid  Liquid  Gas Fluid The basic definition of fluid is that it is a substance which is capable of flowing. Liquids and gases come under the category of fluid. Mechanics Mechanics is the study of force and motion. Fluid Mechanics Fluid Mechanics is the science which deals with the behaviour of fluids at rest and in motion. Hydraulics Hydraulics is the science which deals with the behaviour of water at rest and in motion. Branches of Fluid Mechanics I. Fluid Statics: Fluid statics is the study of fluids at rest. II. Fluid Dynamics: Fluid statistics is the study of fluids in motion. It is classified into two branches.

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Avinas a) Fluid Kinematics: Fluid Kinematics is the study of fluid motion without considering the causes of motion (forces). b) Fluid Kinetics: Fluid Kinetics is the study of fluid motion by considering the causes of motion (forces). Application of Fluid Mechanics  Water distribution and sanitation  Dams  Irrigation  Pumps and Turbines  Water retaining structures  Flood flow analysis  Flow of air in and around buildings  Bridge piers in rivers  Ground-water flow Stress A stress is a force per unit area over which it acts. Stresses have both magnitude and direction, and the direction is relative to the surface on which the stress acts. There are two types of stresses: I. Normal stress: The stress which acts perpendicular to the surface is normal stress. II. Tangential stress: The stress which acts along the surface is tangential stress. Shear stress is tangential stress. Normal stress

Shear stress Strain Strain is the measurement of deformation. In case of fluid, the deformation caused by shear stress is measured in terms of angle, which is known as shear strain. Formal definition of fluid A Fluid is a substance which deforms continuously or flows under the application of shearing forces, however small they may be. For a fluid at rest, there are no shearing forces acting on it, and any force must be acting perpendicular to the fluid. Fluids and solids

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Avinas  Fluids lack the ability of solids to resist deformation.  For a solid, strain is a function of applied stress, provided the elastic limit is not exceeded. For a fluid, the rate of strain is proportional to applied stress.  In a fluid shear strain increases for as long as shear stress is applied. This means the fluid flows as long as the forces acts and will not recover its original position when the force is removed. In a solid shear strain is constant for a fixed shear stress, and if the elastic limit is not exceeded, the deformation disappears when the force is removed. Forces Point force: Single concentrated force Line force: Force which acts along a line  Body force: Body force is the force which acts throughout the volume of body. e.g. gravity (weight), magnetic force, centrifugal force  Surface force: Surface force is the force which acts on the surface of the body. e.g. Pressure, shear force Liquid and Gas  Liquid: incompressible, fixed volume, forms free surface  Gas: compressible, no fixed volume, expand continuously until restrained by a containing vessel, no free surface, fill the vessel in which it is placed. 1.1.2 System and Control volume System: A system is a fixed identifiable quantity of matter. The system boundary separates the boundary from its surroundings. Control volume: A control volume is a fixed region in space through which fluid flows. The region is usually at a fixed location and fixed size. The boundary of the system is its control surface and its shape does not change with time. The element within the control volume obeys the physical laws. This approach makes mathematical analysis simpler. As the fluid flows continuously, only a part of it is considered for analysis. The control volume is chosen arbitrarily for reasons of convenience of analysis. Control surface follows solid boundaries if present. Differential and integral approach  Differential approach: If the control volume is of infinitesimal size, differential equations are used. This approach gives value of variable at a point.  Integral approach: If the control volume is of finite size, integral equations are used. This approach gives global or overall values.

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Avinas 1.1.3 Continuum concept in Fluid Mechanics In Fluid Mechanics, a fluid is considered as a continuous substance. This concept is called continuum concept. In this concept, molecular structure of the fluid is not considered and the separation between molecules is neglected. The fluid properties such as velocity and pressure are a continuous function of space and time. The fluid properties can be considered to be constant at any point in space, which is average of large number of molecules surrounding that point within a characteristic distance. Using continuum concept, the mathematical equations relating the physical laws can be derived easily as we don’t need to consider the motion of individual molecule. This concept is not valid if the mean free path of molecules is greater than the characteristic dimension of fluid considered for analysis. 1.1.4 Velocity profile No slip condition: The velocity of fluid particle immediately in contact with the boundary is same as that of the boundary. This is called no-slip condition. Fixed

Fixed

Fixed

Diagram for velocity profile Velocity of the fluid at the fixed boundary is zero and increases away from the boundary until it reaches a maximum value. If the particles of the fluid move relative to each other with different velocities, shear stress is developed. If the velocity is same, no shear stress is produced. 1.1.5 Basic laws used in Fluid Mechanics Newton’s laws of motion I. A body will remain at rest or in a state of uniform motion in a straight line unless acted upon by an external force. II. The rate of change of momentum of a body is proportional to the force applied and takes place in the direction of action of that force. (Force = mass x acceleration) III. Action and reaction are equal and opposite. Conservation of mass: Mass remains constant. Conservation of momentum (Newton’s second law of motion): Force = rate of change of momentum or F=ma Conservation of energy: Energy remains constant.

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1.2 Fluid properties 1.2.1 Density The density of a fluid is defined as its mass per unit volume.

ρ=

m V

Where ρ = density, m = mass, V = Volume Unit: kg/m3 Dimension: ML-3

ρ decreases with increase of temperature and increases with increase of pressure. As the temperature increases, molecular activity increases and spacing between molecules increases, thus increasing volume and reducing density. If pressure is increased, large number of molecules can be forced into a given volume, thus reducing volume and increasing density. 1.2.2 Specific weight The specific weight of a fluid is defined as its weight per unit volume.

W mg = =ρg V V Where γ = specific weight, W = weight, V = Volume, m = mass, γ=

ρ

= density and g= acceleration

due to gravity γ varies from point to point according to the value of g. Unit: N/m3 Dimension: ML-2T-2

1.2.3 Specific gravity (or relative density) Specific gravity (or relative density) is the ratio of specific weight (or density) of a fluid to that of standard fluid. In case of liquid, the standard fluid is water at 40C.

Sp gr =

Sp wt of fluid Density of fluid ∨ Density of water Sp wt of water

Unit: As it is ratio, it does not have unit. 1.2.4 Specific volume The specific volume of a fluid is defined as its volume per unit mass.

v s=

V 1 = m ρ

Where vs = specific volume, V= Volume, m = mass and

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ρ = density

Avinas Unit: m3/kg Dimension: M-1L3 Variation of temperature and pressure has little effect on density, specific weight and specific volume of liquids as the molecules of liquids are packed together, whereas the impact on these properties in case of gases is significant. 1.2.5 Compressibility and Bulk modulus Compressibility is the change in the volume of fluid under the action of external force. When temperature changes are involved or velocity of flow is very high, the compressibility of a fluid becomes important. It is expressed by Bulk modulus of elasticity If pressure increases from P to P+dP, then the volume V of a given mass will be reduced to V-dV.

Bulk modulus=

Change ∈ pressure Volumetric strain K=

dP −dV /V

(a)

Where K= Bulk modulus of elasticity , dv/v = Volumetric strain -ve sign means decrease in volume with pressure. Unit of K: N/m2 (Pa) Dimension: ML-1T-2

Compressibility is the inverse of the Bulk modulus of elasticity.

Considering unit mass of substance,

V=

1 ρ

Differentiating w.r.t. ρ

dV d (1/ ρ ) = dρ dρ dV −1 = dρ ρ2 −dρ −dρ 1 dV = 2 = . ρ ρ ρ dV = From a and b,

K= ρ

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dP dρ

−dρ .V ρ

(b)

Avinas This shows that the value of K depends on the relationship between the pressure and density. Since density is also affected by temperature, it will depend on how the temperature changes during compression. K increases with increase in pressure and decreases with increase in temperature in case of liquids. This relationship is opposite in case of gases.

1.2.6 Surface tension Surface tension is defined as the tensile force per unit length acting on a line lying in the interface of two fluids. The force is normal to imaginary line in the surface, tangent to the free surface and is same at all points. Surface tension is constant at any given temperature for the surface of the separation of two particular substances but it decreases with increase in temperature because attractive force becomes apparent as the average kinetic energy of molecules increases. Intermolecular attraction is the cause of surface tension. A molecule within the body of a liquid is equally attracted in all directions by the other molecules surrounding it. At the interface between two fluids, the upward and downward attractions are unbalanced, and the surface molecules are pulled inward making the surface like an elastic membrane. The effect of surface tension is to reduce the surface of a free body of a liquid to a minimum (formation of spherical drop). Example of phenomenon of surface tension: raindrops, rise of sap in tree, capillary rise and capillary siphoning, collection of dust particles on water surface Symbol: σ Unit: N/m Dimension: MT-2 a. Pressure intensity inside a droplet σ P σ Consider a small spherical droplet of radius r. Let P be pressure inside a droplet in excess of external pressure and σ be the surface tension. Force due to internal pressure = Force due to surface tension around perimeter

Pxπ r 2=σx 2 πr 2σ P= r Pressure inside soap bubble 7

Avinas (contribution of inside and outside interface)

Force due to internal pressure = Force due to surface tension around perimeter

Pxπ r 2 =2(σx 2 πr ) 4σ P= r Liquid jet (cylindrical)

r

l σ

Force due to internal pressure = Force due to surface tension around perimeter

Pxlx2 r =σx 2 l σ P= r 1.2.7 Capillarity Capillarity is the rise or fall of liquid in a column of very small diameter when the latter is dipped in it. It is caused by surface tension as well as adhesion (attraction between molecules of different substances) and cohesion (attraction between molecules of same liquid). If adhesion is greater than cohesion, the liquid wets the solid and the liquid will rise. If cohesion is greater than adhesion, the liquid does not wet the solid and the liquid will fall. The contact angle is less than 900 for capillary rise and greater than 900 for fall. σ Fall Rise

θ

H H θ

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σ

Let θ is the angle of contact between liquid and solid, d is the diameter of the cylindrical tube, σ is the surface tension and H is capillary rise. As the liquid is at rest, there is no shear stress and therefore no vertical shear forces acting. Weight of the fluid and the vertical component of the surface tension are the only forces acting. Upward pull due to surface tension force= Weight of column contained in height H

σcosθxπd= ρg

π d2 H 4 H=

For water and glass:

4 σcosθ ρgd

or H =

4 σcosθ γd

θ=0

Capillary rise of fluid contained between parallel plates at a distance t t

σ θ

H

H = Capillary rise t = distance between plates σ = Surface tension θ = Angle of contact L= width of plate

Upward force due to surface tension = weight of fluid

( σcosθxL) x 2= ρgHtxL (surface tension force acting on both sides)

H=

2 σcosθ ρgt

Capillary rise between two concentric glass tubes σ

θ

H

r1 r2

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H = Capillary rise σ = Surface tension θ = Angle of contact r 1 = Radius of inner tube r 2 = Radius of outer tube

Avinas

Force due to surface tension =

σcosθx 2 π r 1+ σcosθx 2 π r 2=2 πσcosθ( r 1+ r 2 )

ρg(Volume of fluid contained∈betweentubes)

Weight of fluid contained in height H =

¿ ρg ( π r H −π r H ) = ρgπH ( r −r 2 2

2 1

2 2

2 1

)

Equating

2 πσcosθ ( r 1 +r 2)=ρgπH ( r 2−r 1 ) 2σcosθ H= ρg ( r 2−r 1) 2

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1.2.8 Vapor pressure and cavitation Liquid evaporate because of molecules with sufficient kinetic energy escaping from the liquid surface. The vapor molecules exert a partial pressure in the space, which is called vapor pressure. Vapor pressure depends on temperature and increases with it. In equilibrium, the number of molecules striking the surface and condensing is equal to the number of escaping molecules. When the pressure above a liquid equals the vapor pressure of the liquid, boiling occurs. When flow of liquid passes through a region having pressure less than vapor pressure, there will be local boiling and a cloud of vapor bubbles will form. This phenomenon is known as cavitation. The bubbles of low pressure zone move towards the high pressure zone and collapse under that pressure. If this occurs in contact with a solid surface, serious damage can result. Cavitation can affect the performance of hydraulic machinery such as propellers, turbines and pumps and the impact of collapsing bubbles can cause local erosion of metal surface. 1.2.9 Viscosity Viscosity is the property of a fluid due to which it offers resistance to shear. It is a measure of internal friction which causes resistance to flow. The molecules of gas are not rigidly constrained and cohesive forces are small. So, the molecular mass interchange (momentum) is the cause of viscosity in a gas. As cohesive forces are significant in a liquid, both mass interchange and cohesion contribute to the viscosity of the liquid. Viscosity is practically independent upon pressure and depends on temperature only. If the temperature increases, the molecular interchange will increase. Therefore, the viscosity of a gas will increase with increase in temperature. Cohesion is the predominant cause of viscosity in liquid and since cohesion decreases with temperature, the viscosity of a liquid decreases with increase in temperature. Newton’s law of viscosity

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Avinas Newton’s law of viscosity states that the shear stress is proportional to the rate of deformation or velocity gradient.

du dy du τ =μ dy Where τ τ∝

= shear stress, du/dy = velocity gradient and the constant of proportionality ( coefficient of viscosity. The constant is also called dynamic viscosity or absolute viscosity. Unit of μ : Ns/m2 or Kg/ms or Pa S (in SI) Poise or dyn S/ cm2 (in CGS) 1N = 1 kg m/s2 = 105 Dyn (1Dyn = 1gm cm/s2) 1 NS/m2 = 10 Poise Dimension: ML-1T-1 Derivation

μ¿

=

dx A

A’

D

D’

τ dφ

dy

τ B

C

Let us consider a fluid confined between two plates, where the bottom plate is stationary and the upper plate is moving. Let ABCD is the fluid at any time t. Due to the application of shear force τ, the fluid deforms to A’BCD’ at time t+dt. Let dy = distance between two layers, AA’ = dx and shear strain = dφ. For small angle, dx = dφ. dy Also dx = du. dt Equating

dϕ . dy=du . dt dϕ du = dt dy

(a)

Shear stress is proportional to rate of shear strain.

τ∝

dϕ dt

(b)

From a and b

τ∝ τ =μ

du dy

du dy

Kinematic viscosity is defined as the ratio of dynamic viscosity to density.

υ=

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μ ρ

Avinas Where υ = Kinematic viscosity and ρ = density. Unit of υ : m2/s (SI) Stokes or cm2/s (CGS) 1 m2/s = 104 Stokes Dimension: L2T-1 Relationship between viscosity and temperature Liquids

μ=μ 0

( 1+ αt+1 β t ) 2

where

μ = viscosity of liquid at t 0C in poise,

μ0 = viscosity of liquid at 00C

in poise α, β = constants For gas

μ=μ 0+ αt −β t

2

where

μ = viscosity of gas at t0C in poise,

μ0 = viscosity of gas at 00C in poise

α, β = constants Shear stress in a moving fluid For a fluid at rest, there is no shear stress. When one layer of the fluid moves relative to an adjacent layer, transfer of molecular momentum sets up shear stress which resists the relative motion. The measure of the motion of one layer relative to an adjacent layer is velocity gradient, du/dy. According to Newton’s law of viscosity, shear stress varies linearly with the velocity gradient.

1.3 Classification of fluids a. Newtonian and non-Newtonian fluids Fluids which obey Newton’s law of viscosity are called Newtonian fluids. E.g. water, light oil, air, milk, glycerin, kerosene. For Newtonian fluids, viscosity is constant i.e. viscosity depends on temperature only. Fluids which do not obey Newton’s law of viscosity are called Non-Newtonian fluids. E.g. paint, sewage sludge, crude oil, mud flow. Viscosity is not constant for Non-Newtonian fluids i.e. viscosity depends on temperature, rate of strain and time. b. Compressible and incompressible fluid Fluids whose density changes due to change in pressure are called compressible fluids, e.g. air. Fluids whose density remains constant are called incompressible fluids, e.g. water. b. Ideal and real fluid

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Avinas The fluid which is incompressible and has no viscosity is called ideal fluid (non-viscous or inviscid). It is an imaginary fluid. The fluid which has viscosity is called real fluid (viscous). All the fluids that exist in nature are real fluids.

1.4 Shear stress-rate of strain (velocity gradient) diagram

τ Plastic

Bingahm Plastic Newtonian PseudoPlastic Dilatant

Yield stress rate of strain or velocity gradient

Ideal fluid:

τ =0

Ideal fluid

du Newtonian fluid: τ =μ dy Non-Newtonian fluid:

τ ≠μ

du . The relationship is dy

τ =μ

( ) du dy

n

Classification of Non-Newtonian fluid 

Pseudo-plastic: Viscosity decreases with rate of strain.

τ =μ

( )

du n , n 1 , e.g. printing ink dy
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